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Theorem tsrlin 14328
Description: A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1  |-  X  =  dom  R
Assertion
Ref Expression
tsrlin  |-  ( ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )

Proof of Theorem tsrlin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istsr.1 . . . . 5  |-  X  =  dom  R
21istsr2 14327 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
x R y  \/  y R x ) ) )
32simprbi 450 . . 3  |-  ( R  e.  TosetRel  ->  A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) )
4 breq1 4026 . . . . 5  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
5 breq2 4027 . . . . 5  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
64, 5orbi12d 690 . . . 4  |-  ( x  =  A  ->  (
( x R y  \/  y R x )  <->  ( A R y  \/  y R A ) ) )
7 breq2 4027 . . . . 5  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
8 breq1 4026 . . . . 5  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
97, 8orbi12d 690 . . . 4  |-  ( y  =  B  ->  (
( A R y  \/  y R A )  <->  ( A R B  \/  B R A ) ) )
106, 9rspc2v 2890 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x )  ->  ( A R B  \/  B R A ) ) )
113, 10syl5com 26 . 2  |-  ( R  e.  TosetRel  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) ) )
12113impib 1149 1  |-  ( ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   dom cdm 4689   PosetRelcps 14301    TosetRel ctsr 14302
This theorem is referenced by:  tsrlemax  14329  ordtrest2lem  16933  ordthauslem  17111  ordthaus  17112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-tsr 14307
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